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Geometric rescaling algorithms for submodular function minimization

Dadush, Daniel, Végh, László A. and Zambelli, Giacomo (2020) Geometric rescaling algorithms for submodular function minimization. Mathematics of Operations Research. ISSN 0364-765X (In Press)

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Abstract

We present a new class of polynomial-time algorithms for submodular function minimiza- tion (SFM), as well as a unified framework to obtain strongly polynomial SFM algorithms. Our algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and Fujishige-Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. Firstly, we adapt the geometric rescaling technique, which has recently gained atten- tion in linear programming, to SFM and obtain a weakly polynomial bound O((n4 · EO + n5) log(nL)). Secondly, we exhibit a general combinatorial black-box approach to turn εL-approximate SFM oracles into strongly polynomial exact SFM algorithms. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudo-polynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige-Wolfe algorithm. Combined with the geometric rescaling technique, the black-box approach provides an O((n5 · EO + n6) log2 n) algorithm. Finally, we show that one of the techniques we develop in the paper can also be combined with the cutting-plane method of Lee, Sidford, and Wong [29], yielding a simplified variant of their O(n3 log2 n · EO + n4 logO(1) n) algorithm.

Item Type: Article
Additional Information: © 2020 INFORMS
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 13 Feb 2020 16:09
Last Modified: 04 Jul 2020 18:06
URI: http://eprints.lse.ac.uk/id/eprint/103368

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